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Infinitely More

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The mathematics and philosophy of the infinite. Click to read Infinitely More, by Joel David Hamkins, a Substack publication with tens of thousands of subscribers.

Math

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172 from 2 Jan 2023 to 28 Oct 2026
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weekly (35 in the last year)
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2026 29

  1. What is infinity? —Lectures on Infinity (lecture 7)
  2. Every game of chess ends prematurely
  3. Galileo’s Paradox of Infinity—Lectures on Infinity (lecture 6)
  4. Ordinal definability—how did Gödel do it?
  5. Potential versus Actual Infinity—Lectures on Infinity (lecture 5)
  6. Ordinal definability is definable
  7. Is Infinity Even or Odd?
  8. The Largest Tweetable Number—Lectures on Infinity (lecture 4)
  9. The Paradox of Giants: Strange Consequences in High Dimension—Lectures on Infinity (lecture 3)
  10. Supertasks: Doing Infinitely Many Things — Lectures on Infinity (Lecture 2)
  11. Zeno's Paradox and Infinite Sums—Lectures on Infinity (lecture 1)
  12. Lectures on Infinity
  13. The Natural Field of Ordinals
  14. Fermat’s last theorem in the natural ring of ordinals
  15. Set theory, pluralism, and the multiverse view—About Logic #13
  16. Regrettable Failures in the Natural Ring of Ordinals
  17. The Natural Ring of Ordinals Has Prime Factorization
  18. The Natural Ring of Ordinals
  19. The big bang of numbers
  20. The natural product of ordinals
  21. The Book of Infinity—pre-orders are open
  22. Natural Ordinal Addition
  23. Counting to Epsilon Naught
  24. On the greats and mathematical style
  25. Cantor Normal Form
  26. Indecomposable Ordinals
  27. Ordinal arithmetic
  28. Ultrafinitism as arithmetic potentialism
  29. Anthropomorphizing the Russell paradox

2025 42

  1. Two visions of ultrafinitism intricately intertwined
  2. Ultrafinitism with a largest number
  3. Ultrafinitism
  4. The surreal line is topologically compact—or is it?
  5. The surreal line is topologically disconnected—or is it?
  6. The omnific integers are strange
  7. The omnific integers are an integer part of the surreal numbers
  8. The surreal ω × ω chessboard is bigger than you think
  9. The uncountable transfinite subway
  10. The infinite subway paradox—extending into the transfinite
  11. The infinite subway—a full range of paradox
  12. The infinite subway paradox
  13. The Hilbert program
  14. Tactics versus strategies—the case of chess
  15. The tactical variation of the fundamental theorem
  16. Tactics versus strategies in the theory of games
  17. The curvature of space
  18. Random chess position
  19. Infinite Sudoku
  20. Infinitesimals revisited
  21. Boomerang tilings
  22. Infinite Connect Four
  23. Abstraction in the function concept
  24. On going first
  25. How we might have viewed the continuum hypothesis as a fundamental axiom necessary for mathematics
  26. Take my Philosophy and Logic of Games final exam!
  27. Pushpast
  28. Bubble Monsters
  29. Face Up
  30. The Chocolatier's game
  31. Buckets of Fish!
  32. The fundamental theorem of finite games
  33. The penny partition paradox
  34. The hypergame paradox
  35. Tic tac toe and variants
  36. The Gold Coin game
  37. The game of Nim
  38. Chomp
  39. Twenty one
  40. Finding Fifteen
  41. Interpretability of theories
  42. The zigzag theorem

2024 50

  1. Take my Infinity final exam!
  2. Mutual and bi-interpretation of models
  3. The interpretation translation
  4. Interpretability
  5. Permutations and combinations
  6. Mathematicians disagree on the essential structure of the complex numbers
  7. Representing integers as a sum
  8. Recursive Chess
  9. Finite and transfinite variations of Escape!
  10. Escape!
  11. Throwing Darts at the Real Line
  12. The indispensability of mathematics
  13. What is the infinite?
  14. Tiling problems
  15. We can predict the future
  16. The guessing-box puzzle
  17. The chocolate bar problem
  18. More pointed at than pointing
  19. Infinitary Hat Puzzles and the Aftermath
  20. The Axiom of Choice
  21. Least upper bound principle and continuous induction
  22. Quantifier elimination — Presburger arithmetic
  23. Quantifier elimination — theory of successor
  24. Continuity, instantaneous change, infinitesimals
  25. Quantifier elimination — dense linear orders
  26. Equality as logical
  27. Are the imaginary numbers real?
  28. Reference de re versus de dicto
  29. Definite descriptions
  30. What are the real numbers, really?
  31. Stipulative definitions
  32. Nondefinability and higher definability
  33. Pointwise definability versus Leibnizian discernibility
  34. Famous quotations in first-order predicate logic
  35. Structuralism
  36. Isomorphisms and elementary equivalence
  37. Substructures and chains of models
  38. Theory theory
  39. More applications of induction
  40. A compositional theory of truth
  41. A formal language for first-order predicate logic
  42. Distinguishing mathematical structures by their theories
  43. Chaos train and monkey madness—fun with quantifiers in first-order logic
  44. The countable random graph
  45. An algebra of orders
  46. Mathematical induction
  47. The lattice of all sets of natural numbers
  48. Lattices
  49. The eventual domination order
  50. The surreal numbers

2023 51

  1. Infinitely many primes
  2. Well orders and the ordinal numbers
  3. Linear orders
  4. Order theory—an introduction
  5. Number theory
  6. Functions
  7. Closures of a relation
  8. Equivalence relations
  9. The Liar (a logic song)
  10. Relational logic—an introduction
  11. Interpreting arithmetic
  12. Skolem's paradox
  13. Boolean-algebra logic
  14. Plenitudinous Primes
  15. Logic as algebra, supertruth, superfalsity, paraconsistent logic
  16. Logicism
  17. Fuzzy logic
  18. A classical beginning
  19. Łukasiewicz logic and supervaluation logic
  20. Multi-valued logic—an introduction
  21. Ode to Hippasus
  22. A formal proof system for propositional logic
  23. Compactness for propositional logic
  24. Logical interpolation
  25. Logical duality
  26. A hierarchy of expressive power
  27. The sand reckoner
  28. Expressive completeness of a logical language
  29. Truth functions
  30. A formal language for propositional logic
  31. Classical propositional logic—introduction
  32. Infinite liars
  33. Epistemic logic and the problem of common knowledge
  34. Potential versus actual infinity
  35. The paradox of giants
  36. Self-similar self-similarity
  37. Transfinite recursive constructions
  38. The orders of infinity
  39. The continuum hypothesis
  40. Shadows of infinity cast over the finite realm
  41. How to count
  42. The method of exhaustion
  43. Uncountable infinity
  44. Hilbert's Grand Hotel
  45. Equinumerosity and comparison of size
  46. The largest tweetable number
  47. Supertasks
  48. The infinite coastline paradox
  49. Zeno's paradox
  50. The Book of Numbers
  51. Welcome to Infinitely More